# hyperbolic cases

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## 11—20 of 45 matching pages

##### 11: 36.4 Bifurcation Sets
###### Special Cases
Hyperbolic umbilic bifurcation set (codimension three): … Hyperbolic umbilic cusp line (rib): …
##### 12: 36.2 Catastrophes and Canonical Integrals
###### Normal Forms for Umbilic Catastrophes with Codimension $K=3$
(hyperbolic umbilic).
##### 13: 13.8 Asymptotic Approximations for Large Parameters
Special cases are … For more asymptotic expansions for the cases $b\to\pm\infty$ see Temme (2015, §§10.4 and 22.5)where $w=\operatorname{arccosh}\left(1+(2a)^{-1}x\right)$, and $\beta=\ifrac{(w+\sinh w)}{2}$. …For the case $b>1$ the transformation (13.2.40) can be used. … uniformly with respect to bounded positive values of $x$ in each case. …
##### 14: 19.10 Relations to Other Functions
$\operatorname{arctanh}\left(x/y\right)=xR_{C}\left(y^{2},y^{2}-x^{2}\right),$
$\operatorname{arcsinh}\left(x/y\right)=xR_{C}\left(y^{2}+x^{2},y^{2}\right),$
$\operatorname{arccosh}\left(x/y\right)=(x^{2}-y^{2})^{1/2}R_{C}\left(x^{2},y^{% 2}\right).$
In each case when $y=1$, the quantity multiplying $R_{C}$ supplies the asymptotic behavior of the left-hand side as the left-hand side tends to 0. …
19.10.2 $(\sinh\phi)R_{C}\left(1,{\cosh}^{2}\phi\right)=\operatorname{gd}\left(\phi% \right).$
##### 15: 19.16 Definitions
Just as the elementary function $R_{C}\left(x,y\right)$19.2(iv)) is the degenerate caseAll elliptic integrals of the form (19.2.3) and many multiple integrals, including (19.23.6) and (19.23.6_5), are special cases of a multivariate hypergeometric function …
###### §19.16(iii) Various Cases of $R_{-a}\left(\mathbf{b};\mathbf{z}\right)$
The only cases that are integrals of the third kind are those in which at least one $b_{j}$ is a positive integer. All other elliptic cases are integrals of the second kind. …
##### 16: 14.20 Conical (or Mehler) Functions
14.20.3 $\widehat{\mathsf{Q}}^{-\mu}_{-\frac{1}{2}+\mathrm{i}\tau}\left(x\right)=\frac{% \pi e^{-\tau\pi}\sin\left(\mu\pi\right)\sinh\left(\tau\pi\right)}{2({\cosh}^{2% }\left(\tau\pi\right)-{\sin}^{2}\left(\mu\pi\right))}\mathsf{P}^{-\mu}_{-\frac% {1}{2}+\mathrm{i}\tau}\left(x\right)+\frac{\pi(e^{-\tau\pi}{\cos}^{2}\left(\mu% \pi\right)+\sinh\left(\tau\pi\right))}{2({\cosh}^{2}\left(\tau\pi\right)-{\sin% }^{2}\left(\mu\pi\right))}\mathsf{P}^{-\mu}_{-\frac{1}{2}+\mathrm{i}\tau}\left% (-x\right).$
Special cases:
14.20.13 $P_{-\frac{1}{2}+i\tau}\left(x\right)=\frac{\cosh\left(\tau\pi\right)}{\pi}\int% _{1}^{\infty}\frac{P_{-\frac{1}{2}+i\tau}\left(t\right)}{x+t}\mathrm{d}t,$
14.20.14 $\pi\int_{0}^{\infty}\frac{\tau\tanh\left(\tau\pi\right)}{\cosh\left(\tau\pi% \right)}P_{-\frac{1}{2}+i\tau}\left(x\right)P_{-\frac{1}{2}+i\tau}\left(y% \right)\mathrm{d}\tau=\frac{1}{y+x}.$
For the case of purely imaginary order and argument see Dunster (2013). …
##### 17: 13.24 Series
13.24.3 $\exp\left(-\tfrac{1}{2}z\left(\coth t-\frac{1}{t}\right)\right)\left(\frac{t}{% \sinh t}\right)^{1-2\mu}=\sum_{s=0}^{\infty}p_{s}^{(\mu)}(z)\left(-\frac{t}{z}% \right)^{s}.$
(13.18.8) is a special case of (13.24.1). …
##### 18: Errata
• Section 4.43

The first paragraph has been rewritten to correct reported errors. The new version is reproduced here.

Let $p$ $(\neq 0)$ and $q$ be real constants and

4.43.1
$A=\left(-\tfrac{4}{3}p\right)^{1/2},$
$B=\left(\tfrac{4}{3}p\right)^{1/2}.$

The roots of

4.43.2 $z^{3}+pz+q=0$

are:

1. (a)

$A\sin a$, $A\sin\left(a+\frac{2}{3}\pi\right)$, and $A\sin\left(a+\frac{4}{3}\pi\right)$, with $\sin\left(3a\right)=4q/A^{3}$, when $4p^{3}+27q^{2}\leq 0$.

2. (b)

$A\cosh a$, $A\cosh\left(a+\frac{2}{3}\pi\mathrm{i}\right)$, and $A\cosh\left(a+\frac{4}{3}\pi\mathrm{i}\right)$, with $\cosh\left(3a\right)=-4q/A^{3}$, when $p<0$, $q<0$, and $4p^{3}+27q^{2}>0$.

3. (c)

$B\sinh a$, $B\sinh\left(a+\frac{2}{3}\pi\mathrm{i}\right)$, and $B\sinh\left(a+\frac{4}{3}\pi\mathrm{i}\right)$, with $\sinh\left(3a\right)=-4q/B^{3}$, when $p>0$.

Note that in Case (a) all the roots are real, whereas in Cases (b) and (c) there is one real root and a conjugate pair of complex roots. See also §1.11(iii).

Reported 2014-10-31 by Masataka Urago.

• ##### 19: 22.10 Maclaurin Series
22.10.7 $\operatorname{sn}\left(z,k\right)=\tanh z-\frac{{k^{\prime}}^{2}}{4}(z-\sinh z% \cosh z){\operatorname{sech}}^{2}z+O\left({k^{\prime}}^{4}\right),$
22.10.8 $\operatorname{cn}\left(z,k\right)=\operatorname{sech}z+\frac{{k^{\prime}}^{2}}% {4}(z-\sinh z\cosh z)\tanh z\operatorname{sech}z+O\left({k^{\prime}}^{4}\right),$
22.10.9 $\operatorname{dn}\left(z,k\right)=\operatorname{sech}z+\frac{{k^{\prime}}^{2}}% {4}(z+\sinh z\cosh z)\tanh z\operatorname{sech}z+O\left({k^{\prime}}^{4}\right).$
The radius of convergence is the distance to the origin from the nearest pole in the complex $k$-plane in the case of (22.10.4)–(22.10.6), or complex $k^{\prime}$-plane in the case of (22.10.7)–(22.10.9); see §22.17. …
##### 20: 19.24 Inequalities
Inequalities for $R_{D}\left(0,y,z\right)$ are included as the case $p=z$. … For $x>0$, $y>0$, and $x\neq y$, the complete cases of $R_{F}$ and $R_{G}$ satisfy … Inequalities for $R_{C}\left(x,y\right)$ and $R_{D}\left(x,y,z\right)$ are included as special cases (see (19.16.6) and (19.16.5)). … Special cases with $a=\pm\frac{1}{2}$ are (19.24.8) (because of (19.16.20), (19.16.23)), and …The same reference also gives upper and lower bounds for symmetric integrals in terms of their elementary degenerate cases. …